MATHS – 6.EQUATIONS

Simultaneous equations involving 2 unknowns - x and y are often referred by school students for different methods of solution and more importantly to find whether they are consistent, independent or inconsistent. Let us see 3 simple examples and elaborate on this particular aspect of existence and uniqueness of solution.

1. x+y =2 and x-y =0
2. x+y =2 and 2x+2y = 4
3. x+y =2 and 2x+2y =3

We adopt 2 methods to determine this aspect.

1.Algebraic method

2.Graphic method

I.Independent equations

Test 1 . Algebraic method.

Find the ratios of coefficients of unknowns in the 2 given equations

In ex.1…it is

1/1 =1 for x

and

1/-1 =-1 for y.

the ratios are not equal

IF THE RATIOS ARE DIFFERENT,THEY ARE INDEPENDENT EQUATIONS AND WE GET UNIQUE SOLUTION.

Here we find that x = y = 1 is the one and only one solution.

In ex.2….it is

1 / 2 for x

and

1 / 2 for y

the ratios are equal.

IF THE RATIOS ARE EQUAL THEN THEY ARE DEPENDENT EQUATIONS AND WE CAN HAVE TWO POSSIBILITIES.

A. We may have infinite solutions

We test for this possibility by finding the ratio of constant terms. Here the ratio is 2 / 4 = 1 / 2 which is same as the ratio of coefficients of unknowns.

IF RATIO OF COEFFICIENTS OF UNKNOWNS IS SAME AS THE RATIO OF CONSTANT TERMS THEN THEY ARE DEPENDENT / CONSISTENT AND WE HAVE INFINITE SOLUTIONS.

Here we have x = y = 1 or x = 0 and y =2 or x=2 and y=0 etc..are all possible solutions. In-fact any set of values x = x and y = 2 – x is a solution.

PLEASE NOTE THE COMMENTS UNDER EQUATIONS AND IDENTITIES IN THIS CONTEXT

B. We may have no solution.

IF RATIO OF COEFFICIENTS OF UNKNOWNS IS NOT SAME AS THE RATIO OF CONSTANT TERMS THEN THEY ARE CALLED INCONSISTENT AND WE HAVE NO SOLUTION.

In ex.3..it is

1 / 2 for x

1 / 2 for y

and

2 / 3 for constants which is not equal to the above ratio.

Hence the equations are in-consistent and we have no solution.

Test 2…Graphic method.

Draw graphs of the 2 equations on the same scale on the same paper.

If the 2 graphs intersect , then the point of intersection is the solution.

In Ex.1 we find that the 2 graphs…straight lines in fact intersect
at x = y = 1.Hence they are independent equations and the point of intersection viz.x = y = 1 is the solution.

In Ex.2 …we find that the 2 graphs ….straight lines in fact are coincident. Hence they are dependent/consistent and any point on the line is a solution. Say x = y = 1 or…x=2 and y =0 etc..

In Ex.3...we find that th 2 graphs...straight lines in-fact are parallel and do not intersect.hence, they are inconsistent equations and there is no solution.